Octagon

For other uses, see Octagon (disambiguation).

"Octagonal" redirects here. For other uses, see Octagonal (disambiguation).

Regular octagon
A regular octagon
Type	Regular polygon
Edges and vertices	8
Schläfli symbol	{8} t{4}
Coxeter diagram
Symmetry group	Dihedral (D8), order 2×8
Internal angle (degrees)	135°
Dual polygon	self
Properties	convex, cyclic, equilateral, isogonal, isotoxal

In geometry, an octagon (from the Greek ὀκτάγωνον oktágōnon, "eight angles") is a polygon that has eight sides.

Regular octagon

A regular octagon is a closed figure with sides of the same length and internal angles of the same size. It has eight lines of reflective symmetry and rotational symmetry of order 8. A regular octagon is represented by the Schläfli symbol {8}. The internal angle at each vertex of a regular octagon is 135° and the sum of all the internal angles of any octagon is 1080° (as with all polygons, the external angles total 360°).

Area

The area of a regular octagon of side length a is given by

$A = 2 \cot \frac{\pi}{8} a^2 = 2(1+\sqrt{2})a^2 \simeq 4.828427125\,a^2.$

In terms of the circumradius R, the area is

$A = 4 \sin \frac{\pi}{4} R^2 = 2\sqrt{2}R^2 \simeq 2.828427\,R^2.$

In terms of the apothem r (see also inscribed figure), the area is

$A = 8 \tan \frac{\pi}{8} r^2 = 8(\sqrt{2}-1)r^2 \simeq 3.3137085\,r^2.$

These last two coefficients bracket the value of pi, the area of the unit circle.

The area of a regular octagon can be computed as a truncated square.

The area can also be derived as follows:

$\,\!A=S^2-a^2,$

where S is the span of the octagon, or the second shortest diagonal; and a is the length of one of the sides, or bases. This is easily proven if one takes an octagon, draws a square around the outside (making sure that four of the eight sides touch the four sides of the square) and then taking the corner triangles (these are 45–45–90 triangles) and placing them with right angles pointed inward, forming a square. The edges of this square are each the length of the base.

Given the length of a side a, the span S is:

$S=\frac{a}{\sqrt{2}}+a+\frac{a}{\sqrt{2}}=(1+\sqrt{2})a$

$S \approx 2.414a$

The area is then as above:

$A=((1+\sqrt{2})a)^2-a^2=2(1+\sqrt{2})a^2$

$A \approx 4.828a^2$

Expressed in terms of the span, area is:

$A=2(\sqrt{2}-1)S^2$

$A \approx 0.828S^2$

Another simple formula for the area is

$\ A=2aS.$

More often the span S is known, and the length of the sides, a, is to be determined, as when cutting a square piece of material into a regular octagon. From the above:

$a \approx S/2.414$

The two end lengths e on each side, as well as being e=a/sqrt{2}, may be calculated as:

$\,\!e=(S-a)/2$

Construction and elementary properties

Building a regular octagon via folding of paper sheet

A regular octagon may be constructed as follows:

1. Draw a circle and a diameter AOB, where O is the center and A,B are points on the circumference.
2. Draw another diameter COD, perpendicular to AOB.
3. (Note in passing that A,B,C,D are vertices of a square).
4. Draw the bisectors of the right angles AOC and BOC, making two more diameters EOF and GOH.
5. A,B,C,D,E,F,G,H are the vertices of the octagon.

A Regular Octagon can be constructed through Straightedge and a Compass:

Each side of a regular octagon subtends half a right angle at the centre of the circle which connects its vertices. Its area can thus be computed as the sum of 8 isosceles triangles, leading to the result:

$Area = 2 a^2 (\sqrt{2} + 1)$

for an octagon of side a.

Standard coordinates

The coordinates for the vertices of a regular octagon centered at the origin and with side length 2 are:

• (±1, ±(1+√2))
• (±(1+√2), ±1).

Uses of octagons

The octagonal floor plan, Dome of the Rock.

The octagonal shape is used as a design element in architecture. The Dome of the Rock has a characteristic octagonal plan. The Tower of the Winds in Athens is another example of an octagonal structure. The octagonal plan has also been in church architecture such as St. George's Cathedral, Addis Ababa, Basilica of San Vitale (in Ravenna, Italia), Castel del Monte (Apulia, Italia), Florence Baptistery, Zum Friedefürsten church (Germany) and a number of octagonal churches in Norway. The central space in the Aachen Cathedral, the Carolingian Palatine Chapel, has a regular octagonal floorplan. Uses of octagons in churches also include lesser design elements, such as the octagonal apse of Nidaros Cathedral.

Other uses

• 

  Umbrellas often have an octagonal outline.

• 

  The famous Bukhara rug design incorporates an octagonal "elephant's foot" motif.

• 

  The street & block layout of Barcelona's Eixample district is based on non-regular octagons

• 

  Janggi uses octagonal pieces.

• 

  Japanese lottery machines often have octagonal shape.

• 

  Stop sign used in English-speaking countries, as well as in most European countries

• 

  The trigrams of the Taoist bagua are often arranged octagonally

• 

  Famous octagonal gold cup from the Belitung shipwreck

• 

  Classes at Shimer College are traditionally held around octagonal tables

• 

  The Labyrinth of the Reims Cathedral with a quasi-octagonal shape.

Variations from regular octagons

A vertex-transitive (isogonal) octagon is construct in four mirrors can alternate long and short edges. An edge-transitive octagon (isotoxal) is constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are duals of each other and have half the symmetry order of the regular octagon.

Isogonal

Isotoxal

Skew octagon

A regular skew octagon seen as edges of a square antiprism

A skew octagon is a skew polygon with 8 vertices and edges but not existing on the same plane. The interior of such an octagon is not generally defined. A skew zig-zag octagon has vertices alternating between two parallel planes.

A regular skew octagon is vertex-transitive with equal edge lengths. In 3-dimensions it will be a zig-zag skew octagon and can be seen in the vertices and side edges of a square antiprism with the same D_4d, [2⁺,8] symmetry, order 16.

Derived figures

• 

  The truncated square tiling has 2 octagons around every vertex.
  

• 

  An octagonal prism contains two octagonal faces.
  

• 

  An octagonal antiprism contains two octagonal faces.
  

• 

  The truncated cuboctahedron contains 6 octagonal faces.
  

• 

  The omnitruncated cubic honeycomb
  

Related polytopes

The octagon, as a truncated square, is first in a sequence of truncated hypercubes:

Truncated hypercubes

							...
Octagon	Truncated cube	Truncated tesseract	Truncated 5-cube	Truncated 6-cube	Truncated 7-cube	Truncated 8-cube

As an expanded square, it is also first in a sequence of expanded hypercubes:

Expanded hypercubes

							...
Octagon	Rhombicuboctahedron	Runcinated tesseract	Stericated 5-cube	Pentellated 6-cube	Hexicated 7-cube	Heptellated 8-cube

Petrie polygons

The octagon is the Petrie polygon for these higher-dimensional regular and uniform polytopes, shown in these skew orthogonal projections of in A₇, B₄, and D₅ Coxeter planes.

A7	D5	B4
7-simplex	5-demicube	16-cell	Tesseract

See also

• Bumper pool
• Octagon house
• Octagonal number
• Octagram
• Oktogon, a major intersection in Budapest, Hungary
• Rub el Hizb (also known as Al Quds Star and as Octa Star)
• Smoothed octagon

External links

• Octagon Calculator
• Definition and properties of an octagon With interactive animation
• Weisstein, Eric W., "Octagon", MathWorld.